13-1 designing engineering experiments - ucla …hqxu/stat105/pdf/ch13.pdf · 13-1 designing...

13-1 Designing Engineering Experiments

Every experiment involves a sequence of activities:

1.  Conjecture – the original hypothesis that motivates the experiment.

2.  Experiment – the test performed to investigate the conjecture.

3.  Analysis – the statistical analysis of the data from the experiment.

4.  Conclusion – what has been learned about the original conjecture from the experiment. Often the experiment will lead to a revised conjecture, and a new experiment, and so forth.

13-2 The Completely Randomized Single-Factor Experiment

13-2.1 An Example


13-2.1 An Example

•  The levels of the factor are sometimes called treatments.

•  Each treatment has six observations or replicates.

•  The runs are run in random order.


Figure 13-1 (a) Box plots of hardwood concentration data. (b) Display of the model in Equation 13-1 for the completely randomized single-factor experiment

13-2.1 An Example

13-2 The Completely Randomized Single-Factor Experiment 13-2.2 The Analysis of Variance

Suppose there are a different levels of a single factor that we wish to compare. The levels are sometimes called treatments.


We may describe the observations in Table 13-2 by the linear statistical model:

The model could be written as


Fixed-effects Model

The treatment effects are usually defined as deviations from the overall mean so that:

Also,


We wish to test the hypotheses:

The analysis of variance partitions the total variability into two parts.


The ratio MSTreatments = SSTreatments/(a – 1) is called the mean square for treatments.


The appropriate test statistic is

We would reject H0 if f0 > fα,a-1,a(n-1)


where N=na is the total number of observations.


Analysis of Variance Table


Example 13-1

13-2 The Completely Randomized Single-Factor Experiment Example 13-1


The 95% CI on the mean of the 20% hardwood is


For the hardwood concentration example,


An Unbalanced Experiment


13-2.3 Multiple Comparisons Following the ANOVA

The least significant difference (LSD) is

If the sample sizes are different in each treatment:

13-2 The Completely Randomized Single-Factor Experiment Example 13-2


Example 13-2

Figure 13-2 Results of Fisher’s LSD method in Example 13-2


13-2.4 Residual Analysis and Model Checking



Figure 13-4 Normal probability plot of residuals from the hardwood concentration experiment.



Figure 13-5 Plot of residuals versus factor levels (hardwood concentration).



Figure 13-6 Plot of residuals versus

13-4 Randomized Complete Block Designs

13-4.1 Design and Statistical Analyses

The randomized block design is an extension of the paired t-test to situations where the factor of interest has more than two levels.

Figure 13-9 A randomized complete block design.



For example, consider the situation of Example 10-9, where two different methods were used to predict the shear strength of steel plate girders. Say we use four girders as the experimental units.



General procedure for a randomized complete block design:



The appropriate linear statistical model:

We assume

•  treatments and blocks are initially fixed effects

•  blocks do not interact

• 



We are interested in testing:



The mean squares are:



The expected values of these mean squares are:


Example 13-5


Minitab Output for Example 13-5


13-4.2 Multiple Comparisons

Fisher’s Least Significant Difference for Example 13-5

Figure 13-10 Results of Fisher’s LSD method.



(a) Normal prob. plot of residuals (b) Residuals versus ŷij


(a) Residuals by block. (b) Residuals by treatment

13-1 designing engineering experiments - ucla …hqxu/stat105/pdf/ch13.pdf · 13-1 designing...

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